Consider a time-dependent Lagrangian system with configuration manifold \(Q\)  and time space \(\mathbb {R}\) . In the extended Lagrangian mechanics framework [1]}, we treat time as a dynamic variable and define the extended configuration manifold \(\bar{Q}=\mathbb {R} \times Q\) ; the corresponding state space \(T\bar{Q}\)  is \(\mathbb {R} \times TQ\) . The extended Lagrangian is \(L: \mathbb {R} \times TQ \rightarrow \mathbb {R}\) .
